Highly-oscillatory evolution equations with multiple frequencies: averaging and numerics

Chartier, Philippe; Lemou, Mohammed; Méhats, Florian

doi:10.1007/s00211-016-0864-4

Cited by 11 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(ii) When there are multiple high-frequencies, exponential and trigonometric integrators can sometimes provide accurate simulations, and we refer to [56,55,34,61,49,102,112,48,37,47,54,50,79,90,53,99] (see Section 2.4 for more discussions). An interesting alternative idea is to approximate multiple frequencies by integer multiples of one frequency, which was realized by combining the two-scale reformulation and multi-revolution composition [28]. Another direction of developments was based on the numerical integration of highly oscillatory functions [57,59,58,74], and mainly investigated were the simulations of secondorder equations [66,111,110]; usually one will obtain an iterative method due to implicitness.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Simply improved averaging for coupled oscillators and weakly nonlinear waves

Tao

2019

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed firstorder wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Simply improved averaging for coupled oscillators and weakly nonlinear waves

Tao

2019

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

show abstract

“…Each particle carries its own frequency and now all the particle are coupled to each other through Poisson equation, which as a result mixes all the frequencies. Thus, (2.5) is a multiple-frequency system with a large number N p of degrees [8] and the proposed two-scale formulation is not rigorously working. Here, we give a practical strategy that works well based on our numerical experiments.…”

Section: 3mentioning

confidence: 99%

Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field

et al. 2019

Self Cite

View full text Add to dashboard Cite

In this paper, we consider the numerical solution of highly-oscillatory Vlasov and Vlasov-Poisson equations with non-homogeneous magnetic field. Designed in the spirit of recent uniformly accurate methods, our schemes remain insensitive to the stiffness of the problem, in terms of both accuracy and computational cost. The specific difficulty (and the resulting novelty of our approach) stems from the presence of a non-periodic oscillation, which necessitates a careful ad-hoc reformulation of the equations. Our results are illustrated numerically on several examples.

show abstract

“…Some of these schemes [10] are based on a modulated Fourier expansion of the exact solution [8,14] where the highly oscillatory problem in (2) is reduced to a non-oscillatory limit Schrödinger equation for which any time step restriction is needed. Other schemes are based on multiscale expansions of the exact solution [3,5]. Chartier et al [6] recently introduced a new method which employs an averaging transformation to soften the stiffness of the problem, hence allowing standard schemes to retain their order of convergence.…”

Section: Introductionmentioning

confidence: 99%

High-order uniformly accurate time integrators for semilinear wave equations of Klein-Gordon type in the non-relativistic limit

Mohamad¹,

Oliver²

2020

Preprint

View full text Add to dashboard Cite

We introduce a family of high-order time semi-discretizations for semilinear wave equations of Klein-Gordon type with arbitrary smooth nonlinerities that are uniformly accurate in the non-relativistic limit where the speed of light goes to infinity. Our schemes do not require pre-computations that are specific to the nonlinearity and have no restrictions in step size. Instead, they rely upon a general oscillatory quadrature rule developed in a previous paper (Mohamad and Oliver, arXiv:1909.04616).

show abstract

Highly-oscillatory evolution equations with multiple frequencies: averaging and numerics

Cited by 11 publications

References 22 publications

Simply improved averaging for coupled oscillators and weakly nonlinear waves

Simply improved averaging for coupled oscillators and weakly nonlinear waves

Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field

High-order uniformly accurate time integrators for semilinear wave equations of Klein-Gordon type in the non-relativistic limit

Contact Info

Product

Resources

About